Is the Motion of a Rubber Ball in a Concave Surface Simple Harmonic Motion?

[vkash]

Homework Statement

lets have a look on scene;
there is a concave of radius R.(like concave lens) there is a rubber ball of radius r(r<<R). It is placed in equilibrium position in the concave. Now it is displaced by small distance along concave and released,friction is sufficient enough for pure rolling of ball. What kind of motion will it did.

Homework Equations

all equations i know.

The Attempt at a Solution

let me say it's acceleration at any instant is a.net frictional force acting on the ball is f.
so our equation of motion comes out to be.
ma=mg*sin(θ)-f (here θ is the angle made by the center on the concave with line joining it with equilibrium position)
θ is too small so we can write sin(θ)=~θ
equation changes to
a=gθ-f/m
since it is pure rolling angular acceleration(α) = linear acceleration(a)/radius of ball(r)
so α=a/r
torque(T)=2/5mr²*α=2/5mr²a/r (taking around center of ball)
T=(frictional force)fr=2/5mr²a/r
from here we got the frictional=(2/5)ma
putting these values in first equation which was a=gθ-f/m
we got a+2/5a=gθ
simplifying it
a=5/7g θ
a=Ar=5/7gθ (A is angular acceleration with center of concave body with line joining it with mean position line)
A=5/7gθ/r
A directly proportional to θ.
hence this is equation of angular SHM.
this is my self generated question so can't tell what is it's answer in book. So it's u who will tell me either this answer is correct or incorrect?
Have i did any mistake in applying Torque equations?(i am not good in rotational dynamics)
If i am wrong somewhere then please tell me.I will thankful to u.
thanks!

 

[SammyS]

All that the fiction accomplishes is that it causes the ball to roll, apparently without slipping. This will not dissipate any energy.

 

[vkash]

All that the fiction accomplishes is that it causes the ball to roll, apparently without slipping. This will not dissipate any energy.

If anything doesn't dissipate energy does it mean that it is SHM. I think no. You need to prove it to confirm.

 

[SammyS]

If anything doesn't dissipate energy does it mean that it is SHM. I think no. You need to prove it to confirm.

I was merely making a statement to clarify the role of friction in this situation.

I have not stated that it is SHM, nor have I stated that it's not SHM.

However, I will say here, that most small oscillation problems, like this one, are SHM.

 

[asteriatic]

I am unable to provide a definitive answer without conducting further experiments and analysis on the system described. However, based on the equations and reasoning provided, it appears that the motion of the rubber ball in the concave is indeed simple harmonic motion. This is supported by the fact that the acceleration is proportional to the displacement, which is a defining characteristic of SHM. Additionally, the fact that the ball is rolling without slipping and the frictional force is sufficient for pure rolling also supports the idea of SHM. However, further analysis and experimentation may be needed to fully confirm this conclusion.